In a school courtyard in Lucknow on a dusty Sunday afternoon, the final push in a heroic campaign to drive a crippling disease from the planet is under way. Among scores of wide-eyed children, four-year-old Mohamed Yusuf is brought to the big wooden table under the yellow banners by his mother Afsar Jahan. Uncomprehending but compliant, he tilts his head back and opens his mouth to receive two drops of polio vaccine. His less fortunate sister Saba Banu, 12, comes across the open space to join them, strikingly beautiful in her bright blue sari, swinging her deformed limb this way and that on her crutches. Saba's right leg is stunted from polio, which she contracted when she was two.

This campaign in the most densely populated state of India is intended to stop polio blighting other lives as it has Saba's. Nobody knows how long it will last, how much more effort will be required or whether, in the end, we will get there at all.

In this country of desperate poverty and large families, disabled infants can be left in the rubbish or face a lifetime of begging on the street, but Afsar Jahan will not allow that to happen to Saba. "She has always gone to school," she says of her daughter. "I will give her the best education I can so she will be compensated." Like every other parent, she would like Saba to marry but she knows her daughter's prospects are damaged. Afsar Jahan helps spread the word about immunisation in her community. "I have suffered," she says. "Now I tell everyone, 'Please, do not make the same mistake.'"

The Lucknow schoolyard is on the frontline in the war against a virus that regularly used to maim children in Britain. Calliper and crutches were a common sight in the 1950s, when the UK had 45,000 cases. The arrival of the polio vaccine in the 1960s wiped out the disease in developed countries and triggered a remarkable aspiration – to eradicate it from the world. The job was supposed to have been finished at the turn of the millennium, but nearly a decade and $7bn on, polio eludes us still. Last year, there were 1,500 cases in the world – a tiny fraction of the 350,000 in 1985, but a real and present danger not only in India, Pakistan, Nigeria and Afghanistan, where polio is still endemic, but also to other countries where migrants and travellers can so easily take it. The numbers have hardly shifted in five years. But can it be defeated now? In 1979, the World Health Organisation (WHO) declared smallpox eradicated. But 30 years on, polio presents a different set of problems.

Fetid grey rivers bubble the length of every street in Lucknow on their sluggish way to the river Gomti, carrying all kinds of domestic waste. In summer, the monsoon rains flood the streets, spreading raw sewage. Polio thrives in human faeces. Small children, dirty hands to mouths, are most at risk. The target of the eradication campaign is the under-fives.

Beside a major traffic junction, the clamour of car horns assaulting the ears, a cluster of middle-aged Britons in canary-yellow polo shirts bob about, waving and shouting at families in rickshaws with young children. In their hands they have droppers containing vaccine and pens to mark the little finger of the left hand of every immunised child. These are some of a group of 86 Rotarians from all over the UK who have flown in to help with the latest mass immunisation day in India's two remaining endemic states – Uttar Pradesh, of which Lucknow is the capital, and Bihar – as well as Delhi, where children are at risk from migration from both areas. Now and again a motorbike pulls up, a whole family on board, and the Rotarians race to reach a small child perched on the machine and squeeze drops in his mouth. They hand out whistles and pens and pull cardboard masks over bemused children's faces.

India is the key to a polio-free world, says Oliver Rosenbauer of the WHO's polio eradication initiative. In Nigeria, all the state governors committed to polio eradication at the start of this year. Kano, in the north, was the global polio epicentre, but there have been no cases of type 1 virus [the worst] for six months. Now India, Rosenbauer says, is "very, very close" to eradicating type 1 polio. Type 2 has already disappeared.

But there are huge challenges. "The quality of the immunisation campaign is very high," Rosenbauer says. "They reach upwards of 95% of kids. But there are half a million babies born in Uttar Pradesh every month, extremely poor sanitation and a tropical weather system that helps transmission."

The basic work now is wearisome. Every child in the endemic states must be immunised again and again. There are no records. No child can be crossed off a list. Every couple of months an immunisation day is scheduled by the government and all-out efforts are made to give the vaccine drops to 800,000 under-fives. Most will have been dosed five or six times a year. There is a danger that polio fatigue will set in.

"A lot of Indian Rotarians are sick to their gills by these frequent rounds," Deepak Kapoor, chairman of Rotary's all-India polio committee tells the British visitors. Their arrival will boost the morale of the Indian club members, whose work now is not vaccination, which the government has largely taken over, but endless awareness and mobilisation campaigns, organising rallies of children bearing home-made placards, recruiting celebrity cricketers to the cause.

This is why the Brits are in canary yellow. They are here to be noticed. Mike and Bernice Yates, who are leading the tour, have a business in Ahmedabad and have been coming on polio trips for six or seven years. "We have different clothes on and attract attention, especially in the villages," says Bernice. "We bring out a lot of people out of curiosity."

Down Lucknow's narrow residential streets, houses are marked and re-marked on their walls or doors in biblical style with what looks like an algebraic equation. "P" means protected from polio – any children within have been immunised. A number above a line denotes the number of the house in the street to have been checked. The number below identifies the team seeking out small children. An arrow shows which way the team went next.

As time has gone on, fewer and fewer families have brought their children to the 2,709 vaccination "booths" in the city. Most now wait for the immunisers to come to them in the five-day house-to-house hunt that follows.

But some are hard to find. Indian Rotarians talk of HRAs (high risk areas) – the brick kilns where migrant populations work, and the slums. But, jokes Ajay Saxeena of Rotary India's national PolioPlus committee, there is now another type of HRA – high rise apartments, the lofty dwelling places of Lucknow's rising middle class. They have had their children immunised. They don't see why they need to keep on doing it. At the vast, turreted, British-built railway station at the city's heart, where teams of yellow-vested government immunisers leap out to accost parents of small children, some of the more affluent just brush them away.

There are some 200 brick kilns in four areas on the outskirts of Lucknow, each marked by a towering chimney. Women sit on the ground, scooping mud from a watery pit, patting and rolling it into large rugby balls. Men press the mud balls into moulds and push out the brick shapes, marked with the name of the kiln on the top, which are sun-dried and fired. A man carries 24 bricks at a time on a yoke. A woman walks quickly, with 10 stacked on a piece of wood on her head and unloads them in pairs, reaching above her head with practised hands. Her face is covered in brick dust.

Their windowless houses could be mistaken for stacks of bricks, with just a curtain for a door. In summer, the kilns close for the rains and the migrant workforce trek back to their villages in one of the poorest parts of India. There is a high risk that the polio virus will trek with them.

Outside, a mother is cooking while her three-year-old hides in the cool semi-darkness. The little boy, Abishek Chohan, was given polio drops two months ago. If all goes well, a polio team will call again this week. The last visit to Jagdish Brick Field is recorded on the outside wall of the manager's office – "14/09/09". Three more babies have arrived since then – one has just been born and is lying under a blanket next to its exhausted mother on a wooden bed in a lightless brick hut.

But the next factory, Sunil Brick Field, appears to have missed out. People shake their heads. "No polio service," they say. These families are from Hamirpur, 200km away in the Himalayan foothills. Shanthi is three, Kajal is 18 months and Sadena is one year old. None of them has had the polio vaccine, the community says. No drops. There are no marks on the manager's office wall. Migrant workers like these are easy to miss.

Not so the people of the slums. The rag pickers' children are going nowhere. They live among the detritus of the city, their shacks made of sticks, sacking and the plastic bags their parents and older siblings collect all day long for packing and reselling. These are immigrant Muslims from Bangladesh and Assam. If they are happy today to allow their children to swallow the polio drops, it is largely thanks to an impressive Islamic scholar and leader, Khalid Rasheed, president of the Ulema Council of India.

"There were a number of misconceptions in the Muslim community about the disease: that there was a conspiracy on the part of the Americans and foreign powers, that this vaccine would make them impotent and infertile, so Muslim parents were not giving it to their children," says Rasheed at his madrassa in Lucknow, where a vaccination booth has been set up.

So a conference was called of Muslim scholars, who consulted Muslim doctors, and a consensus was reached that the vaccine, which has been universally given in Islamic states such as Iran, Iraq and Saudi Arabia, did no harm. It's a message Rasheed and other Islamic leaders in India now promote. "We have been able to give all this information to people who are generally illiterate and have no knowledge of what the vaccine is," he says. They use everything from newspaper adverts to appeals during Friday prayers to reach Muslim families, who tend to be among the poorest communities. About a year ago, 70% of polio cases were among Muslims, who are only 30% of the population. Now, 30% of cases are in Muslim families. India's Islamic scholars plan a trip to Nigeria in 2010 to urge Muslims there to form a polio vaccine promotion committee, as they have in India.

Some of the British Rotarians, who have all paid their own passage, are stunned by what they see. The group includes businessmen, a judge, teachers and GPs. The Rotarians started it all in 1979, when Rotary International, based in Evanston, Chicago, linked up with the government of the Philippines to immunise its under-fives. Since then, through fetes, coffee mornings, donations and tin-shaking all over the well-to-do world, Rotary has raised $896m (£545m).

Most of the volunteers here have not been to India before. All will take back moving accounts of the work Rotary is involved in and urge those back home to keep the faith, even though the eradication target has been missed and missed again.

"It's been extremely worthwhile. I spent an hour last night writing a journal. I found it difficult to put into words what it meant," says Steve Martin from Merseyside, who trains police dogs to sniff out explosives.

They are amazed by the scale of the Indian eradication effort but also by the complexity of the problem. But Mike Yates says he is a lot more optimistic than he was two years ago. "I have been here three times and each time the city is improving," he says. Free housing is now being built to replace some slums.

The Rotarians will return to Britain to tour the clubs with their pictures and tell members why they must keep giving. It is a hard message, nearly a decade after the job should have been finished, and there are fears that funding from the G8 may be slipping. But, they will no doubt tell their colleagues, it just needs one more last push.